A non-iterative straightening algorithm and orthogonality for skew Schur modules

TL;DR

This paper introduces a non-iterative straightening algorithm for skew Schur modules, providing explicit formulas and an orthogonal basis, advancing the algebraic understanding of these modules.

Contribution

It generalizes Fulton's determinantal construction to skew shapes and develops a non-iterative straightening algorithm with an orthogonal basis for skew Schur modules.

Findings

Explicit non-iterative straightening algorithm for skew Schur modules

Identification of a natural inner product via Gram-Schmidt orthogonalization

Construction of an orthogonal basis (D-basis) for skew Schur modules

Abstract

We generalize Fulton's determinantal construction of Schur modules to the skew setting, providing an explicit and functorial presentation using only elementary linear algebra and determinantal identities, in parallel with the partition case. Building on the non-iterative straightening formula of the first author for partition shapes, we develop a non-iterative straightening algorithm for skew Schur modules that expresses arbitrary elements in a new D-basis with an explicit closed coefficient formula. We then show that this D-basis is the result of applying Gram-Schmidt orthogonalization to the semistandard tableau basis, which identifies a natural inner product on the skew Schur module and recasts straightening as an orthogonal projection.